Question: Factor the following expression: $8$ $x^2+$ $9$ $x$ $-14$
Answer: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(8)}{(-14)} &=& -112 \\ {a} + {b} &=& & & {9} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-112$ and add them together. Remember, since $-112$ is negative, one of the factors must be negative. The factors that add up to ${9}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-7}$ and ${b}$ is ${16}$ $ \begin{eqnarray} {ab} &=& ({-7})({16}) &=& -112 \\ {a} + {b} &=& {-7} + {16} &=& 9 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {8}x^2 {-7}x +{16}x {-14} $ Group the terms so that there is a common factor in each group: $ ({8}x^2 {-7}x) + ({16}x {-14}) $ Factor out the common factors: $ x(8x - 7) + 2(8x - 7) $ Notice how $(8x - 7)$ has become a common factor. Factor this out to find the answer. $(8x - 7)(x + 2)$